Local Brownian motions

In this thesis, stochastic processes closely related to a prominent process called Brownian motion is introduced. The processes are called local Brownian motions. Two rich families are constructed, which can be used to model stochastic dynamical systems in such fields as finance, biology, and physics. Stochastic calculus with respect to local Brownian motion is developed, and strong solutions are presented for some stochastic differential equations driven by the noise generated by local Brownian motions.